Curry-Howard-Lambek correspondence
From HaskellWiki
Uchchwhash (Talk | contribs) m (inhibit -> inhabit) |
m |
||
(6 intermediate revisions by 5 users not shown) | |||
Line 1: | Line 1: | ||
[[Category:Theoretical foundations]] |
[[Category:Theoretical foundations]] |
||
+ | |||
+ | {{Foundations infobox}} |
||
The '''Curry-Howard-Lambek correspondance''' is a three way isomorphism between types (in programming languages), propositions (in logic) and objects of a Cartesian closed [[Category theory|category]]. Interestingly, the isomorphism maps programs (functions in Haskell) to (constructive) proofs in logic (and ''vice versa''). |
The '''Curry-Howard-Lambek correspondance''' is a three way isomorphism between types (in programming languages), propositions (in logic) and objects of a Cartesian closed [[Category theory|category]]. Interestingly, the isomorphism maps programs (functions in Haskell) to (constructive) proofs in logic (and ''vice versa''). |
||
__TOC__ |
__TOC__ |
||
− | == The Answer == |
+ | == Life, the Universe and Everything == |
As is well established by now, |
As is well established by now, |
||
<haskell>theAnswer :: Integer |
<haskell>theAnswer :: Integer |
||
Line 29: | Line 31: | ||
p2pShare n | n == 0 = Warning "Share! Freeloading hurts your peers." |
p2pShare n | n == 0 = Warning "Share! Freeloading hurts your peers." |
||
| n < 0 = Error "You cannot possibly share a negative number of files!" |
| n < 0 = Error "You cannot possibly share a negative number of files!" |
||
− | | n > 0 = OK ("You are sharing " ++ show n ++ " files." |
+ | | n > 0 = OK ("You are sharing " ++ show n ++ " files.") |
</haskell> |
</haskell> |
||
So any one of <hask>OK String</hask>, <hask>Warning String</hask> or <hask>Error String</hask> proves the proposition <hask>Message String</hask>, leaving out any two constructors would not invalidate the program. At the same time, a proof of <hask>Message String</hask> can be pattern matched against the constructors to see which one it proves. |
So any one of <hask>OK String</hask>, <hask>Warning String</hask> or <hask>Error String</hask> proves the proposition <hask>Message String</hask>, leaving out any two constructors would not invalidate the program. At the same time, a proof of <hask>Message String</hask> can be pattern matched against the constructors to see which one it proves. |
||
Line 55: | Line 57: | ||
The type is, actually, <hask>forall a b c. (a -> b) -> (b -> c) -> (a -> c)</hask>, to be a bit verbose, which says, logically speaking, for all propositions <hask>a, b</hask> and <hask>c</hask>, if from <hask>a</hask>, <hask>b</hask> can be proven, and if from <hask>b</hask>, <hask>c</hask> can be proven, then from <hask>a</hask>, <hask>c</hask> can be proven (the program says how to go about proving: just compose the given proofs!) |
The type is, actually, <hask>forall a b c. (a -> b) -> (b -> c) -> (a -> c)</hask>, to be a bit verbose, which says, logically speaking, for all propositions <hask>a, b</hask> and <hask>c</hask>, if from <hask>a</hask>, <hask>b</hask> can be proven, and if from <hask>b</hask>, <hask>c</hask> can be proven, then from <hask>a</hask>, <hask>c</hask> can be proven (the program says how to go about proving: just compose the given proofs!) |
||
+ | |||
+ | == Negation == |
||
+ | Of course, there's not much you can do with just truth. <hask>forall b. a -> b</hask> says that given <hask>a</hask>, we can infer anything. Therefore we will take <hask>forall b. a -> b</hask> as meaning <hask>not a</hask>. Given this, we can prove several more of the axioms of logic. |
||
+ | |||
+ | <haskell> |
||
+ | type Not x = (forall a. x -> a) |
||
+ | |||
+ | doubleNegation :: x -> Not (Not x) |
||
+ | doubleNegation k pr = pr k |
||
+ | |||
+ | contraPositive :: (a -> b) -> (Not b -> Not a) |
||
+ | contraPositive fun denyb showa = denyb (fun showa) |
||
+ | |||
+ | deMorganI :: (Not a, Not b) -> Not (Either a b) |
||
+ | deMorganI (na, _) (Left a) = na a |
||
+ | deMorganI (_, nb) (Right b) = nb b |
||
+ | |||
+ | deMorganII :: Either (Not a) (Not b) -> Not (a,b) |
||
+ | deMorganII (Left na) (a, _) = na a |
||
+ | deMorganII (Right nb) (_, b) = nb b |
||
+ | </haskell> |
||
== Type classes == |
== Type classes == |
||
Line 66: | Line 89: | ||
means, logically, there is a type <hask>a</hask> for which the type <hask>a -> a -> Bool</hask> is inhabited, or, from <hask>a</hask> it can be proved that <hask>a -> a -> Bool</hask> (the class promises two different proofs for this, having names <hask>==</hask> and <hask>/=</hask>). |
means, logically, there is a type <hask>a</hask> for which the type <hask>a -> a -> Bool</hask> is inhabited, or, from <hask>a</hask> it can be proved that <hask>a -> a -> Bool</hask> (the class promises two different proofs for this, having names <hask>==</hask> and <hask>/=</hask>). |
||
− | This proposition is of existential nature (not to be confused with [[existential type]]). A proof for this proposition (that there is a type that conforms to the specification) is (obviously) a set of proofs of the advertized proposition (an implementation), by an <hask>instance</hask> |
+ | This proposition is of existential nature (not to be confused with [[existential type]]). A proof for this proposition (that there is a type that conforms to the specification) is (obviously) a set of proofs of the advertised proposition (an implementation), by an <hask>instance</hask> |
declaration: |
declaration: |
||
Line 78: | Line 101: | ||
</haskell> |
</haskell> |
||
− | A not-so-efficient Quicksort implementation would be: |
+ | A not-so-efficient sort implementation would be: |
<haskell> |
<haskell> |
||
− | quickSort [] = [] |
+ | sort [] = [] |
− | quickSort (x : xs) = quickSort lower ++ [x] ++ quickSort higher |
+ | sort (x : xs) = sort lower ++ [x] ++ sort higher |
− | where lower = filter (<= x) xs |
+ | where (lower,higher) = partition (< x) xs |
− | higher = filter (> x) xs |
||
</haskell> |
</haskell> |
||
− | Haskell infers its type to be <hask>forall a. (Ord a) => [a] -> [a]</hask>. It means, if a type <hask>a</hask> satisfies the proposition about propositions <hask>Ord</hask> (that is, has an ordering defined, as is necessary for comparison), then <hask>quickSort</hask> is a proof of <hask>[a] -> [a]</hask>. For this to work, somewhere, it should be proved (that is, the comparison functions defined) that <hask>Ord a</hask> is true. |
+ | Haskell infers its type to be <hask>forall a. (Ord a) => [a] -> [a]</hask>. It means, if a type <hask>a</hask> satisfies the proposition about propositions <hask>Ord</hask> (that is, has an ordering defined, as is necessary for comparison), then <hask>sort</hask> is a proof of <hask>[a] -> [a]</hask>. For this to work, somewhere, it should be proved (that is, the comparison functions defined) that <hask>Ord a</hask> is true. |
== Multi-parameter type classes == |
== Multi-parameter type classes == |
||
− | Haskell makes frequent use of multiparameter type classes. Type classes use a logic language (Prolog-like), and for multiparamter type classes they define a relation between types. |
+ | Haskell makes frequent use of multiparameter type classes. Type classes constitute a Prolog-like logic language, and multiparameter type classes define a relation between types. |
=== [[Functional dependencies]] === |
=== [[Functional dependencies]] === |
||
These type level functions are set-theoretic. That is, <hask> class TypeClass a b | a -> b</hask> defines a relation between types <hask>a</hask> and <hask>b</hask>, and requires that there would not be different instances of <hask>TypeClass a b</hask> and <hask>TypeClass a c</hask> for different <hask>b</hask> and <hask>c</hask>, so that, essentially, <hask>b</hask> can be inferred as soon as <hask>a</hask> is known. This is precisely functions as relations as prescribed by set theory. |
These type level functions are set-theoretic. That is, <hask> class TypeClass a b | a -> b</hask> defines a relation between types <hask>a</hask> and <hask>b</hask>, and requires that there would not be different instances of <hask>TypeClass a b</hask> and <hask>TypeClass a c</hask> for different <hask>b</hask> and <hask>c</hask>, so that, essentially, <hask>b</hask> can be inferred as soon as <hask>a</hask> is known. This is precisely functions as relations as prescribed by set theory. |
Latest revision as of 21:35, 21 February 2010
Haskell theoretical foundations
General: Lambda calculus: Other: |
The Curry-Howard-Lambek correspondance is a three way isomorphism between types (in programming languages), propositions (in logic) and objects of a Cartesian closed category. Interestingly, the isomorphism maps programs (functions in Haskell) to (constructive) proofs in logic (and vice versa).
Contents |
[edit] 1 Life, the Universe and Everything
As is well established by now,
theAnswer :: Integer theAnswer = 42
[edit] 2 Inference
A (non-trivial) Haskell function maps a value (of typerepresentation :: Bool -> Integer representation False = 0 representation True = 1
[edit] 3 Connectives
Of course, atomic propositions contribute little towards knowledge, and the Haskell type system incorporates the logical connectives and , though heavily disguised. Haskell handles conjuction in the manner described by Intuitionistic Logic. When a program has type , the value returned itself indicates which one. The algebraic data types in Haskell has a tag on each alternative, the constructor, to indicate the injections:
data Message a = OK a | Warning a | Error a p2pShare :: Integer -> Message String p2pShare n | n == 0 = Warning "Share! Freeloading hurts your peers." | n < 0 = Error "You cannot possibly share a negative number of files!" | n > 0 = OK ("You are sharing " ++ show n ++ " files.")
show :: Message String -> String show (OK s) = s show (Warning s) = "Warning: " ++ s show (Error s) = "ERROR! " ++ s
The conjuction is handled via an isomorphism in Closed Cartesian Categories in general (Haskell types belong to this category): . That is, instead of a function from to Z, we can have a function that takes an argument of type X and returns another function of type , that is, a function that takes Y to give (finally) a result of type Z: this technique is (known as currying) logically means .
(insert quasi-funny example here)
So in Haskell, currying takes care of the connective. Logically, a proof of is a pair (a,b) of proofs of the propositions. In Haskell, to have the final C value, values of both A and B have to be supplied (in turn) to the (curried) function.
[edit] 4 Theorems for free!
Things get interesting when polymorphism comes in. The composition operator in Haskell proves a very simple theorem.
(.) :: (a -> b) -> (b -> c) -> (a -> c) (.) f g x = f (g x)
[edit] 5 Negation
Of course, there's not much you can do with just truth.type Not x = (forall a. x -> a) doubleNegation :: x -> Not (Not x) doubleNegation k pr = pr k contraPositive :: (a -> b) -> (Not b -> Not a) contraPositive fun denyb showa = denyb (fun showa) deMorganI :: (Not a, Not b) -> Not (Either a b) deMorganI (na, _) (Left a) = na a deMorganI (_, nb) (Right b) = nb b deMorganII :: Either (Not a) (Not b) -> Not (a,b) deMorganII (Left na) (a, _) = na a deMorganII (Right nb) (_, b) = nb b
[edit] 6 Type classes
A type class in Haskell is a proposition about a type.
class Eq a where (==) :: a -> a -> Bool (/=) :: a -> a -> Bool
declaration:
instance Eq Bool where True == True = True False == False = True _ == _ = False (/=) a b = not (a == b)
A not-so-efficient sort implementation would be:
sort [] = [] sort (x : xs) = sort lower ++ [x] ++ sort higher where (lower,higher) = partition (< x) xs
[edit] 7 Multi-parameter type classes
Haskell makes frequent use of multiparameter type classes. Type classes constitute a Prolog-like logic language, and multiparameter type classes define a relation between types.
[edit] 7.1 Functional dependencies
These type level functions are set-theoretic. That is,[edit] 8 Indexed types
(please someone complete this, should be quite interesting, I have no idea what it should look like logically)